Co-Dimension-Two Grazing Bifurcations In Impact Oscillators With Symmetrical Constraints

Impacting phenomena exist in a large number of mechanical systems. The collision introduces an essential non-linearity and discontinuity. One of the most special phenomena, occurring in impact systems, is the so-called grazing, where the impact velocity of oscillators tends to zero. Because of the grazing behavior, the vibor-impact systems can exhibit complicated dynamic behaviors. The co-dimension-two grazing bifurcations in impact oscillators with symmetrical constraints are investigated in this thesis, and the main research contents and research results are as follows:Firstly, for a class of n-degree-of-freedom impact oscillator with symmetrical constraints, using a classical approach of discontinuity mapping, the Poincare map of double grazing periodic motion is obtained by combination of discontinuity maps and smooth Poincare maps. The Poincare mapping obtained above is used to discuss the stability of double grazing periodic motion, and the expressions of co-dimension-two grazing bifurcation in four different cases are derived respectively. Finally, the expressions are simplified further by the Jacobian matrices of smooth Poincare maps.Secondly, a two-degree-of-freedom impact oscillator and a three-degree-of-freedom impact oscillator with symmetrical constraints are investigated respectively. From the initial conditions and periodicity conditions, the criterion for existence of a double grazing periodic motion is presented and verified by numerical simulation. The schematics of co-dimension-two grazing bifurcation points and grazing curve are described by numerical simulation. Then, the dynamical features near critical points of grazing co-dimension-two bifurcation are analyzed, and the existence regions of different types of periodic-impact motions are obtained. Furthermore, the phase diagrams and the unfolding chart of the co-dimension-two grazing bifurcations are given numerically.